This month's topics:

Confidence, Women and Calculus I

A research article in PLoS ONE (July 13, 2016; authors Jessica Ellis, Bailey Fosdick, Chris Rasmussen) says it all in the title: "Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit." This large study (2266 students at 129 2- and 4-year colleges and universities), run under the auspices of the MAA, focused on students taking Calculus I, i.e. the first-semester "mainstream" course that is usually part of a STEM career. They report the factor of 1.5 mentioned in the title, state "These results show that Calculus I is a critical 'leak' in the STEM pipeline, especially for women" and that "only targeting efforts at college calculus and beyond would increase the number of women entering the STEM workforce by 75%," and then proceed to ask why this happens.

"The students who were not going on to Calculus II were given a list of potential reasons and asked to select all that resonated with them. ... The proportions of students who cited each reason were comparable across men and women, except for one:

'I do not believe I understand the ideas of Calculus I well enough to take Calculus II.'"

Roughly twice as many women as men chose this as one of their reasons. "However, previous research suggests that this perceived lack of understanding among women is not because women do not actually understand the material as well as men; on the contrary, a meta-analysis of gender differences in mathematics found no differences in ability and a study specifically looking at gender differences in Calculus I found that women outperform men."

Where does this "perceived lack of understanding" develop? The PLoS report includes two diagrams that show how women and men lose confidence at about the same rate during their first term, but that women start college with a significantly lower mathematical confidence level than men.

Change in confidence for 'capable' students. In these graphs, the vertical axis is "Mean Confidence." They show the change in students' responses between two surveys, one (Pre) before the students had taken Calculus I and one (Post) after. For analysis, the students were in two groups: STEM Intending: those considering majors in the physical sciences or engineering, and STEM Interested: those considering majors (e.g. pre-med) which required some calculus. Persisters are the students who chose to go on to Calculus II, switchers are those who did not. Images from Ellis et al.

"This work points to female students' mathematical confidence entering college as a major contributing factor to women's participation in the STEM workforce, and thus more work is needed to understand the factors (such as classroom environment, home environment, extra curricular involvement, etc.,) that help to shape students' perceptions of their own success before they enter college. Such work is outside the scope of the current study, but our work indicates that significant efforts should be aimed at targeting such questions."

The PLoS study was the subject of a report by Maggie Kuo in Science, July 22.

Update on Smale's 7th problem

Lisa Zyga posted "Researchers chip away at Smale's 7th unsolved roblem in mathematics" on Phys.org, July 15, 2016. As she explains, the underlying problem is Thomson's Problem (J. J. Thomson, 1904): "how to arrange equal charges (such as electrons) on the surface of a sphere in a way that minimizes their electrostatic potential energy--the energy caused by all of the electrons repelling each other."

Some actual and putative solutions to Thomson's problem. According to Zyga (see Beltrán), the problem "has been rigorously solved only for numbers of 2, 3, 4, 6, and 12 charges." Note that for 4, 6, and 12 the charges are at the vertices of a platonic solid. Image adapted from Sloane and Hardin, as shown here.

As Beltrán puts it, "This beautiful problem is terribly challenging!" The main method of solution for $N$ charges is to explore the roughly $N^2$-dimensional graph of the energy function $V$ on the space of all configurations, looking for a minimum. Part of the challenge is that the number of local minima grows exponentially with $N$. Back in 1998 Steve Smale gave a quantitative formulation to the matter by asking for an algorithm that could pick a set of exploration starting points where $V$ was close (within a general constant times $\log N$) to the global minimum value; this is the seventh of his "Mathematical Problems for the next century" (Math. Intelligencer20,2).

The progress Zyga refers to is research reported July 6 by Dhagash Mehta and coauthors: "Kinetic Transition Networks for the Thomson Problem and Smale's 7th Problem." What Mehta et al. discovered is that the Thomson problem may not be so untractable, practically speaking. They constructed disconnectivity graphs for $N = 132, 135, 138, 141, 144, 147,$ and $150$ where the vertical coordinate is energy, the vertices are local minima and two are connected by an edge if there is a single transition state (a connection between pairs of minima via a steepest-descent path) between them. They found that these graphs have a "palm tree" organization.

The disconnectivity graph for $N=150$. Note the "palm tree" structure. Two of the local minima are illustrated by their configurations, where each charge corresponds to a polygon; most charges (green) have 6 nearest neighbors, but some have 5 (red) or 7 (blue). An important feature of the underlying network is that the maximum number of steps (edges) between vertices (local minima) is small. In fact the maximum is 7 for $N=150$ (it is 6 for 147 and 5 for the other values of $N$ the authors studied): the local minima live in a "small world". In particular for $N=150$ any local minimum is at most 7 steps from the global minimum. Image courtesy of Halim Kusumaatmaja, one of the co-authors.

Zyga quotes Mehta: "In this work, methods developed by the theoretical chemistry community have helped understand a deep mathematical problem. Often it is the other way around." On the other hand, Zyga: "As the researchers explain, it's easier to solve Thomson's problem in these particular cases than it is to solve Smale's problem (of choosing good starting points). So although the results will likely be useful, they do not go very far toward solving Smale's seventh problem." She quotes David Wales, another co-author: "I think 'chip away' is about right."

As the latest collective stab at enlightenment, Castelvecchi tells us, "Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University's Research Institute for Mathematical Sciences (RIMS)." Castelvecchi spoke with the number theorist Kiran Kedlaya (UCSD) and relays that "Although at first Mochizuki's papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae, experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial. ... Still, Kedlaya says that the more he delves into the proof, the longer he thinks it will take to reach a consensus on whether it is correct. He used to think that the issue would be resolved perhaps by 2017. 'Now I'm thinking at least three years from now.'" Some other opinions: Jeffrey Lagarias (Michigan) "says that he got far enough to see that Mochizukis' work is worth the effort: 'It has some revolutionary new ideas.'" And from Vesselin Dimitrov (Yale): "The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me. Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature."

A more positive spin comes from Jacob Aron, reporting in New Scientist, August 2, 2016: "Mathematicians finally starting to understand epic ABC proof." Aron's main source was Ivan Fesenko (Nottingham), one of the organizers of the meeting of which he says "It definitely went better than expected." An important part was the participation by Mochizuki himself: "It was the key part of the meeting. He was climbing the summit of his theory, and pulling other participants with him, holding their hands." Fesenko adds: "I expect that at least 100 of the most important open problems in number theory will be solved using Mochizuki's theory and further development."

Cahal Milmo's report for iNews, "Unveiled: Maths solution so hard only four mathematicians in the world understand it (and it took them four years)," seems to have some new information: "A statement issued after the Kyoto meeting said the IUT [Mochizuki's Inter-universal Teichmüller Theory] papers had been 'thoroughly studied and verified in their entirety by at least four mathematicians.' It continued: 'These papers are currently being refereed, and, although they have not yet been officially accepted for publication, the refereeing process is proceeding in an orderly, constructive, and positive manner.'"